A Quantum Computational Approach to Correspondence Problems on Point Sets

Vladislav Golyanik Christian TheobaltMax Planck Institute for Informatics, Saarland Informatics Campus

Abstract

Modern adiabatic quantum computers (AQC) are al-ready used to solve difficult combinatorial optimisationproblems in various domains of science. Currently, onlya few applications of AQC in computer vision have beendemonstrated. We review AQC and derive a new algorithmfor correspondence problems on point sets suitable for ex-ecution on AQC. Our algorithm has a subquadratic com-putational complexity of the state preparation. Examplesof successful transformation estimation and point set align-ment by simulated sampling are shown in the systematic ex-perimental evaluation. Finally, we analyse the differencesin the solutions and the corresponding energy values.

1. Introduction

Since their proposal in the early eighties [8, 43, 27],quantum computers have attracted much attention of physi-cists and computer scientists. Impressive advances bothin quantum computing hardware and algorithms have beendemonstrated over the last thirty years [40, 30, 61, 58, 42,19, 25, 49, 65, 48]. Quantum computers are not universallyfaster than conventional machines, but they can natively ex-ecute algorithms relying on quantum parallelism, i.e., theability to perform operations on exponentially many super-imposed memory states simultaneously [59].

To harness the advantages, carefully designed algorithmsare required. Nowadays, the motivation to take advantage ofquantum effects in computing is also facilitated by the clas-sical computing paradigm approaching its limits, since thequantum effects are becoming non-neglectable while man-ufacturing and using conventional CPUs. As a result, al-ternative paradigms such as massively parallel computingdevices have been brought into being.

While universal gate quantum computer technology hasnot yet reached the maturity, modern adiabatic quantum an-nealers (AQA) are already capable of solving difficult real-world combinatorial optimisation problems [15, 14, 23, 49].The primary difference of universal gate quantum comput-ing and AQA is that the latter can address objectives formu-lated as quadratic unconstrained binary optimisation prob-

Figure 1: Different 2D point sets — fish [47], qubit, kanji and composer— aligned with our QA approach. For every pair of point sets, the initialmisalignment is shown on the left, and the registration is shown on theright. QA is the first transformation estimation and point set alignmentmethod which can be executed on adiabatic quantum computers.

lems (QUBOP) defined as

arg minq∈Bn

qTPq, (1)

where q is a set of n binary variables, and P is a symmetricmatrix of weights between the variables. The operationalprinciple of AQA is grounded on the adiabatic theorem ofquantum mechanics [17] which states that

if a quantum-mechanical system is in the ground stateof a time-dependent Hamiltonian and parameters ofthis Hamiltonian are changing gradually enough, thesystem will continue to remain in the ground state dur-ing the evolution (see Table 1 for quantum notions).

(2)

In their seminal paper, Farhi et al. [26] have shownthat the adiabatic principle (2) can be used for solvingNP-complete optimisation problems and laid the founda-tion for adiabatic quantum computing. Several years later,Aharonov et al. [2] theoretically showed the equivalence be-tween classical quantum computing and quantum anneal-ing models. As of 2019-2020, general-purpose quantumcomputers accessible for research purposes and applicationscontain up to 20 qubits [19]. In contrast, the latest quantumannealers support up to 210 qubits [21]1. Nevertheless, due

1the amount of logical qubits which are available on this system is anorder of magnitude lower since most qubits are used for the error correction

1

to design and practical restrictions, quantum algorithms forthe gates model such as Shor’s prime number factorisation[61] or Grover’s search algorithms [30] cannot be imple-mented on current quantum annealers.Motivation and Contributions. Considering recent suc-cessful applications of AQA in several fields of computa-tional science [42, 49, 65], we are motivated to investigatehow useful AQA can be for computer vision and whichproblems can be potentially solved on the new hardware.The vast majority of available materials about quantum an-nealers are either oriented to physicists or lack technical de-tails and clarity. Our goal is to fill this gap, introduce thereader into the modern AQA and provide all notions andthe background to understand, analyse, simulate and designquantum algorithms for computer vision which can poten-tially run on modern AQA, as well as interpret the results.

We consider correspondence problems on point setswhich have various applications in computer vision. Theyconsist in finding an optimal rigid transformation betweeninputs [34, 13, 47, 66]. While transformation estimation as-sumes known matches, point set alignment is more generaland targets, in addition, the recovery of correspondences.We consider two inputs, i.e., a fixed reference point set and atemplate undergoing a rigid transformation. Thus, our goalis to design a quantum approach for point set alignmentwhich can potentially run on AQA and show that it offersadvantages compared to the classical counterparts.

Therefore, we adapt the recent progress in rigid pointset alignment and formulate a globally multiply-linked en-ergy functional which does not require any intermediatecorrespondence updates [29]. In the gravitational approach(GA) [29], the optimal alignment is achieved when the grav-itational potential energy (GPE) of the system with two in-teracting particle swarms is locally minimal. Proceedingfrom GA, we build the weight matrix P for the associatedQUBOP (1) which is unalterably valid in the course of theoptimisation. Along with that, we are targeting at a methodwhich is implementable on classical hardware and can solvereal-world problems, cf. Fig. 1. To summarise, the maincontributions of this paper are:

• A self-contained and detailed introduction into modernquantum annealers for computer vision problems, in-cluding notions from quantum physics and computing(Sec. 2), modern adiabatic quantum annealers (Sec. 3)including D-WAVE (Sec. 3.2), and previous and relatedworks from quantum computing (Sec. 4).

• The first quantum approach (QA) to transformation esti-mation (Sec. 5) and point set alignment (Sec. 6) whichcan run on the upcoming quantum annealers (Sec. 6.2).

• Experimental analysis of the proposed method in a simu-lated environment on several datasets (Sec. 7).

quantum notion classical counterpartqubit (states |0〉 and |1〉) bit (states 0 and 1)

(time-dependent) Hamiltonian energy functionaleigenstate some energy state

ground state globally optimal energy statequantum system evolution optimisation processquantum annealing [26] simulated annealing [39]

Table 1: Quantum notions and their counterparts in computer vision.

2. Preliminaries, Definitions and Notations

In this section, we introduce the reader into the basics ofquantum computing. See Table 1 for a lookup of notionsspecific to AQA which have counterparts and interpretationin the classical optimisation theory for computer vision.Qubit. Quantum computing encompasses tasks which canbe performed on quantum-mechanical systems [53]. Quan-tum superposition and entanglement are two forms of par-allelism evidenced in quantum computers. A qubit is aquantum-mechanical equivalent of a classical bit. A qubit|φ〉 — written in the Dirac notation — can be in the state|0〉, |1〉 or an arbitrary superposition of both states denotedby |φ〉 = α|0〉 + β|1〉, where α and β are the (generally,complex) probability amplitudes satisfying |α|2 + |β|2 = 1.In quantum computing, the state |0〉+|1〉√

2denoted by |+〉 is

often used for initialisation of a qubit register. The state ofa qubit remains hidden during the entire computation andreveals when measured. If qubits are entangled, measur-ing one of them influences the measurement outcome of theother one [59]. During the measurement, the qubit’s stateirreversibly collapses to one of the basis states |0〉 or |1〉.Efficient physical realisation of a qubit demand very lowtemperatures. Otherwise, thermal fluctuations will destroyit and lead to arbitrary changes of the measured qubit state.

One possible physical implementation of a qubit is anelectron which possesses a spin, i.e., its intrinsic magneticmoment [53, 63]. The spin of an electron can be manip-ulated and brought to the state spin down, spin up, or asuperposition of both. A concrete experimentally realisedscheme that uses this property is represented by an atomof phosphorus 31P embedded into a 28Si silicon lattice at-tached to a transistor [37, 46, 67]. The nucleus of 31P hasa positive charge compensated by electrons. The bundle ofelectrons in the transistor is filled up to the energetic levelbetween the energy of spin-down and spin-up state of 31P.To change a state of a 31P–28Si qubit, a microwave pulse ofthe frequency — which is equal to the resonance frequencyof the atom — is applied to it. The new state |φ〉 depends onthe duration of the exposure. A transistor is used to measurea state of the 31P–28Si qubit. If the extra electron of 31P tun-nels into the electron bundle, a positive charge is measuredin the transistor indicating the spin-up state (e.g., |1〉).

Fig. 2-(a) visualises a qubit with a so-called Blochsphere. Every qubit can be both in a superposition and en-

2

x

y

z + + + + ... +

...

...

1 1 0 1 0...

s = 0:

s = 1:

annealing

...

...

  [0;

1]s

+

(a) (b)

Figure 2: (a): Schematic depiction of a qubit with a Bloch sphere. Spin-up or |1〉 is located on the north pole, and spin down or |0〉 is located onthe south pole. The state |0〉+|1〉√

2with equal probability amplitudes to

measure |1〉 and |0〉 values is geodetically equidistant to both poles. Apoint on the surface of the Bloch sphere corresponds to a valid pure state|φ〉 = α|0〉 + β|1〉. (b): Schematic visualisation of adiabatic quantumannealing (AQA). At the beginning, all qubits are initialised in the state|+〉. After the annealing is finished, the qubit states are measured andreturned. After the measurement, the states of variables are classical.

tangled with other qubits. Thus, quantum superposition isthe property that calculations are performed on all possibleinputs simultaneously which can result in exponential par-allelism in the number of qubits. When entangled, states ofqubits cannot be described independently from each other.Schrodinger Equation. In the universal or gates model,changes are expressed by a series of unitary transformationsapplied to qubits. This is a useful practical simplification,while the evolution of every quantum-mechanical systemcan be described more precisely by continuous Schrodingerequation, which in common notation reads:

− i ddt|φ(t)〉 = H(t) |φ(t)〉. (3)

For the simplicity, we denote here by |φ〉 a state of n qubitsat time t, and H(t) is a Hamiltonian which is, in this case,a 2n×2n Hermitian matrix. Thus, a discrete time evolutionof the quantum system is given by a unitary transformation.Hamiltonian. Hamiltonian H is an energy operator of asystem of n qubits. It defines the energy spectrum of asystem or, in our case, the space of all possible solutions.The ground state of the system is its lowest energy eigen-state. Finding a ground state of a Hamiltonian is equivalentto finding an optimal solution to the problem. The expec-tation value of Hamiltonian 〈H〉 provides an instantaneousenergy of a given qubit configuration. In correspondenceproblems, 〈H〉 is a quantitative characteristic of point setalignment. We denote by ∆(H) the spectral gap of H, i.e.,the difference between the energies of the ground state andthe second lowest eigenstate. The spectral gap influencesthe annealing rate and is considerable for algorithm designand evaluation in quantum annealing.Pauli Matrices. An arbitrary Hamiltonian of a n-qubit-system can be expressed by a linear combination of tensorproducts of Pauli matrices denoted by:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

). (4)

The Pauli matrices are 2×2 Hermitian and unitary. Togetherwith the identity σ0 = I2×2, they form a basis for C2×2.σx flips the probabilities to measure |0〉 and |1〉, whereasσz|0〉 = |0〉, and σz|1〉 = −|1〉.Pseudo-Boolean Functions. A pseudo-boolean function isa real vector-valued function of n boolean variables denotedby x of the form F(x) : Bn → RM, where M is the numberof real-valued outputs.Quantum Annealing. Quantum annealing is a heuristiccombinatorial optimisation method for finding global op-tima which relies on quantum effects (superposition, entan-glement and tunnelling) [11, 36]. In particular, it is used tofind a ground state of an Ising Hamiltonian [35, 57], whichencodes the target computational problem, see Fig. 2-(b).

Quantum annealing is the quantum counterpart of sim-ulated annealing [45, 39]. Starting from the superpositionstate [|+〉]⊗n (this is a shorthand for n qubits in the state|+〉, cf. (9)), the system evolves according to (3) under anexternal time-dependent magnetic field (a transverse field).When the external field is faded away, the system reachesthe ground state of an Ising model [35]. According to (2), ifan external magnetic field is changing gradually enough, thesystem remains near the ground state with high probabilitythroughout the optimisation. Quantum annealing systemstaking advantage of (2) are called adiabatic quantum com-puters (AQC). QUBOP is the most common problem formwhich can be mapped to current realisations of AQC.

3. Modern Adiabatic Quantum ComputationAdiabatic quantum computation is a form of quantum

annealing which relies on the adiabatic theorem of quantummechanics (2) [17]. Starting from a ground state of an ini-tial default Hamiltonian HI , an AQC system adiabaticallyevolves into the ground state of a problem Hamiltonian HP

which encodes a solution to a problem [26]. In the case ofadiabatic quantum annealing (AQA), the problem Hamil-tonian HP is given by the Ising model [35]:

HP =∑j∈V

hjσzj +

∑(j, k)∈EP

Jj,k σzj ⊗ σzk, (5)

with the Kronecker product ⊗, hj denoting exterior localmagnetic fields and Ji,j standing for the pairwise connec-tions between the particles. V is a set of particles, and EPis a set of edges (intra-particle links) of the graph. Eq. (5) iswritten in a notation common in physics. The first term of(5) on the right side in the explicit notation reads

Hj∈VP =

[σz ⊗ I⊗ . . .⊗ I

][I⊗ σz ⊗ . . .⊗ I

]. . .[

I⊗ I⊗ . . .⊗ σz]T

2n×n2n

h1 I2n×2n

h2 I2n×2n

...hn I2n×2n

n2n×2n

,

(6)

3

where I without a subscript is a 2 × 2 identity matrix. The

second term H(j, k)∈EP

P of (5) can be expressed in a similarmanner, involving pairs of σz in the tensor product depend-ing on the connectivity of the lattice.

Theoretically, each particle can interact with any otherparticle from the whole set of qubits. In practice, the cou-plings are restricted to local neighbourhoods (see Sec. 3.2).Thus, (5) describes a system of N interacting spin-½ parti-cles under the influence of distributed magnetic forces, andin the expanded form, HP is a 2n × 2n matrix. Finding aground state of an Ising model is an NP-hard problem [7].In the ground state, the spin configuration of all particleswhich minimises Ising energy EIsing is given by:

EIsing =∑i

hisi +∑i,j

Ji,jsisj , (7)

where si ∈ {1,−1} denotes two possible spin measurementoutcomes of a spin-½ particle.

3.1. Quantum System Evolution

Solving NP-hard problems such as QUBOP on a clas-sical computer requires exponential time in the size of theinput. The main idea of the AQC is that a QUBOP (1) canbe mapped to the Ising model (5) and optimised by allowingthe system to evolve according to the adiabatic principle (2).Once annealing is finished, the qubit register will representthe solution to the programmed problem with a high proba-bility [26] (cf. Appendix A on the annealing rate criterion).The initial Hamiltonian of the system is always initialisedin the state

HI = −∑j∈V

Bxσxj , (8)

where Bx > 0 stands for a magnetic field pointing in the xdirection. The ground state of (8) is a symmetrised superpo-sition with equal normalised probability amplitudes for thestates |0〉 and |1〉 for all qubits, i.e.,

[|+〉]⊗n =

(|0〉+ |1〉√

2n

)⊗n. (9)

This initial state (9) is comparably easy to construct by radi-ating a microwave of the same duration and wavelength toall qubits. In mathematical terms, (9) is obtained by apply-ing a Hadamard transform H = 1√

2

[1 11 −1

]to n |0〉 qubits.

The lowest energyEGS = −nBx of (8) is achieved whenall qubits in the system point in the anti-parallel direction ofthe magnetic field, so that σxj |sj〉 = |sj〉. During AQC, theinitial Hamiltonian HI is evolving into the problem Hamil-tonian HP , with a high probability of reaching the groundstate of HP [26]. The interpolation between the Hamilto-nians can be written as

H = [1− s] HI + s HP , (10)

with s ∈ [0; 1] being the time in relative units from the startof annealing at s = 0 until reaching the ground state of HP

at s = 1. The problem Hamiltonian and the final state of thesystem depend on the objective function f(x) or the matrixof weights between the qubits P in (1). After the annealingis accomplished, the state of each qubit is measured, andthe result corresponds to the solution of the programmedproblem with a high probability. At this stage, the states ofall binary variables are classical, and not quantum anymore.

To remain in the ground state during the system evolu-tion, the annealing rate has to be carefully chosen. The con-dition of adiabaticity (2) is derived from the time-dependentperturbation theory of quantum systems. It is achievedwhen the average energy pumped into the system per timeinterval T is smaller than the minimal energy difference be-tween the ground state and the first excited state. This state-ment was quantified in [4] which generalises the originaladiabatic theorem [17] for periodic driving, see Appendix Afor further details.

3.2. Quantum Annealer D-WAVE

D-WAVE relies on the adiabatic criterion in its specifiedform and currently supports up to ≈2000 qubits [22]. It re-flects the state of the art in physical realisation of quantumprocessors. It is relatively inexpensive to bring the system inthe superposition state, and every computation on D-WAVEstarts with the problem-independent HI (8). Qubits can in-teract with a restricted number of other qubits, and it is pos-sible to define qubit equality and entanglement constraints[22]. Possible interactions can be seen from the chimeragraph which schematically depicts the layout of the quan-tum processor [22, 16]. At the same time, the physicallyrealised connectivity can model QUBOP with arbitrary con-nectivities through an internal conversion [16]. The draw-back is that in the worst case, a quadratic increase in thenumber of variables is required. A fully connected graph oflayers with N qubits would require N2 qubits for process-ing. Some QUBOP cannot be mapped to the chimera graph,and some problems can be mapped in multiple ways [55].

4. Previous and Related WorkUniversal Quantum Computers. The paradigm of the uni-versal quantum computer originates in the attempts to gaincontrol over individual quantum systems in the early eight-ies [62, 53]. Later, extending the control to multiple quan-tum systems has attracted the interest of physicists, promis-ing to facilitate discoveries in quantum physics [53]. By thattime, it was noticed that simulating a quantum-mechanicalsystem on a classical computer requires exponential timein the number of simulated elements [43, 27]. “Can youdo it2 with a new kind of computer – a quantum com-

2to simulate quantum-mechanical effects

4

puter?” [27] is a famous quote by R. Feynman which hastriggered research on quantum computers in the subsequentyears. The so-called no-cloning theorem [56, 64] belongs tothe first discoveries strongly influenced quantum informa-tion theory and quantum computations. Nowadays, quan-tum computers can be used not only to fulfil their primarygoal, i.e., to simulate quantum-mechanical systems for dif-ferent branches of science, but also to solve other computa-tional problems — such as balanced function decision prob-lem [24], quantum Turing machines for complexity analy-sis [12], prime number factorisation and discrete logarithms[61], database search [30], graph matching [3], data classifi-cation [58] and principal component analysis [42] — fasterthan on classical machines. The related field of quantumcommunication and quantum key distribution has alreadyfound broad practical use nowadays [10, 9, 60].Classical Methods using Quantum Analogies. Quantum-mechanical effects inspired multiple techniques for conven-tional computers including variants of genetic and evolu-tionary algorithms [31, 32], non-rigid mesh analysis [5] andimage segmentation [6], among others.Quantum Annealers in Computer Vision. Only a fewtheoretical results and applications of AQC to image pro-cessing, machine learning and computer vision are known.Neven et al. [51] have shown how image recognition canbe formulated as QUBOP. Image classification on 12 × 12images with AQC was addressed in [52]. The approach ofO’Malley et al. can learn facial features and reproduce fa-cial image collections [54]. Boyda et al. [18] propose anAQC method to detect areas with trees from aerial images.Several methods target classification, dimensionality reduc-tion and training of deep neural networks [50, 38, 1]. Not alltheoretical findings of these works are possible to test on thereal AQC hardware yet. Nonetheless, we believe that it isessential to explore the theory and highlight the advantagesof the upcoming hardware for computer vision tasks.

5. Quantum Transformation EstimationIn this section, we introduce our QA to transformation

estimation. The inputs are a reference point set [xn] ∈X ∈ RD×N and a template point set [yn] ∈ Y ∈ RD×N ,n ∈ {1, . . . , N}. N is the number of points in both pointsets and D is the dimensionality of the points. We assumethat translation is resolved, the centroids of the point setscoincide, and points are in correspondence.

5.1. Transformation Estimation in 2D

To obtain an advantage in solving transformation estima-tion on a quantum annealer we should avoid uniform sam-pling of rotations applied to Y. Elements of the rotationgroup are non-commutative, and it is not possible to formu-late multiplication of basis rotations as QUBOP. Instead, wepropose to represent the transformation matrix as a linear

combination of basis elements. Recall that for any rotationmatrix, R−1 = RT. Rotation in 2D consists of four ele-

ments, i.e., R =

(r1,2 r2,2r2,1 r2,2

). Additively, we can create a

basis for all possible values of R and encode the influenceof the additive elements as binary variables. Consider in-stead the power series of R in 2D. Every such matrix has acorresponding skew-symmetric matrix of the form

S = θM, M =

[0 −11 0

], (11)

with a real number θ. According to the Cayley-Hamiltontheorem, S2 + θ2I = 0 which leads to the following expo-nential map for R with power series:

R = exp(S) =

cos(θ) I +

(sin(θ)

θ

)S = cos(θ) I + sin(θ) M.

(12)

From (12) we see that R is composed of an identityweighted by cos(θ) and M weighted by sin(θ). If the basiswould resemble additive elements I and M of the exponen-tial map, we can stronger constrain the resulting R. We seethat r1,1 is entangled with r2,2, and r1,2 is entangled withr2,1. Eventually, we need fewer basis elements, the optimi-sation will finish faster and the method can be also imple-mented and tested on a classical computer. Thus, our basisQ = {Qk} for R is a compound of K = 20 elements:{

Qk = ωC ∈ R2×2,∀ω ∈ {0.5, 0.2, 0.1, 0.1, 0.05},∀C ∈ {I,M,−I,−M}

}.

(13)

Since we want to find R which minimises the distances be-tween the corresponding points (xn,yn), we multiply eachtemplate point with a negative sign −yn with each basiselement Qk and stack the result into Φ:

Φ =

xT1 xT

2 . . . xTN

−[Q1y1]T −[Q1y2]

T . . . −[Q1yN ]T

−[Q2y1]T −[Q2y2]

T . . . −[Q2yN ]T

......

. . ....

−[QKy1]T −[QKy2]

T . . . −[QKyN ]T

. (14)

Next, we set the weight matrix in (1) as

P = ΦΦT, (15)

and the final QUBOP reads

arg minq∈B21

qTΦΦTq. (16)

In total, 21 qubits are required to resolve the transformationon AQC in 2D, with the first qubit of q being fixed to |1〉.After solving (16) with quantum annealing and measuringq, we obtain a classical bitstring q. The resulting (perhapsapproximate) R is then obtained by unembedding as

R =

K∑k=1

qk+1Qk. (17)

5

5.2. Transformation Estimation in 3D

In 3D, a skew-symmetric matrix can be represented as

S = θM, M =

m1,1 m1,2 m1,3

m2,1 m2,2 m2,3

m3,1 m3,2 m3,3

=

0 a b−a 0 c−b −c 0

,(18)

where θ, a, b and c are real numbers, and a2 + b2 + c2 =1. In the 3D case, the Cayley-Hamilton theorem states that−S3 − θ2S = 0. The exponential map for R in 3D withpower series reads

R = exp(S) =I +

(sin θ

θ

)S +

(1− cos θ

θ2

)S2 =

I + sin θM + (1− cos θ) M2.(19)

Next, M can be decomposed as follows:

M = a

0 1 0−1 0 00 0 0

︸ ︷︷ ︸

Ma

+ b

0 0 10 0 0−1 0 0

︸ ︷︷ ︸

Mb

+ c

0 0 00 0 10 −1 0

︸ ︷︷ ︸

Mc

.

(20)Regarding M, we see that

• {m1,2;m2,1}, {m1,3;m3,1} and {m2,3;m3,2} are mu-tually dependent or entangled,

• mi,j ∈ [−1; 1],

• M = −MT, Ma = −MTa , Mb = −MT

b and Mc =−MT

c , i.e., they are anti-symmetric, and

• M2 =

[v−1 d ed v−2 fe f v−3

]is symmetric negative semi-

definite, with {v−1 , v−2 , v

−3 } ∈ R−, and {d, e, f} ∈ R.

The basis for rotation in 3D is comprised of the identitymatrix I, Ma, Mb, Mc as well as the basis for M2:

Md =

0 1 01 0 00 0 0

,Me =

0 0 10 0 01 0 0

,Mf =

0 0 00 0 10 1 0

.

(21)Thus, our basis Q3D = {Q3D

k } for R in 3D is a compoundof K = 80 elements:{

Q3Dk = ωC3D ∈ R3×3,∀ω ∈ {0.5, 0.2, 0.1, 0.1, 0.05},

∀C3D ∈ {I,−I, Ma,−Ma,Mb,−Mb,Mc,−Mc,

Md,−Md,Me,−Me,Mf ,−Mf}}.

(22)The final QUBOP and the unembedding (i.e., decoding thesolution to QUBOP) after quantum annealing for the 3Dcase are obtained similarly to (14)–(17) with q ∈ B81 (q0

remains fixed to |1〉 and Qk are replaced by Q3Dk in (17)).

6. Quantum Point Set RegistrationIn point set registration, the input point sets are of dif-

ferent cardinalities, and correspondences between pointsare, generally, not known, i.e., [xn] ∈ X ∈ RD×N and[ym] ∈ Y ∈ RD×M , m ∈ {1, . . . ,M}. N and M arethe numbers of points in the reference and template, respec-tively, while D is the point dimensionality. The objective ofpoint set alignment is to recover rotation R (R−1 = RT,det(R) = 1) and translation t aligning Y to X. We assumethat the translation is resolved in the pre-processing step bybringing the point set centroids into coincidence.

Point set alignment can be alternatingly solved on AQCby finding some point matches and estimating the transfor-mation with the given correspondences in the ICP fashion[13]. This would result in a sequence of QUBOP of the form(16). To express alignment as a single QUBOP, we have tofind an energy functional which is correspondence-free andwhich, when minimised in one shot on AQC, would resultin an optimal alignment. The desired form of the energyfunctional has been recently shown in the literature [29].

6.1. Particle Dynamics Based Alignment

Barnes-Hut Rigid Gravitational Approach (BHRGA)[29] is a recent point set alignment method with a singleenergy functional which remains unchanged during the en-tire optimisation. BHRGA is a globally multiply-linked ap-proach, i.e., all ym interact with all xn. In [29], point setsare aligned by minimising the mutual gravitational poten-tial energy (GPE) E of the corresponding system of parti-cles in the force field induced by X:

E(R, t) =∑m

∑n

µymµxn‖R ym + t− xn‖2 , (23)

where µymand µxn

denote masses of ym and xn, respec-tively. With no imposed boundary conditions, particles areinitialised with unit masses. In [29], (23) is optimised withthe Levenberg-Marquardt algorithm [41, 44], and the op-timum is achieved when the system’s GPE is locally min-imal. Without acceleration by a 2D-tree, the method hasquadratic complexity and (23) involves all possible interac-tions between the template and reference points.

We can now derive a QUBOP in the similar fashion asin Sec. 5 for the transformation estimation. Note, however,that the bases (13) and (22) allow for affine transformationsand scaling. Thus, implicitly, we would optimise

E(R, t, s) =∑m

∑n

µymµxn‖R yms+ t− xn‖2 , (24)

where the scalar s is the scaling of the template. As provenin [28], allowing for scale in globally multiply-linked pointset alignment results in the shrinkage of the template to asingle point with a very high probability. To remedy the

6

problem, either prior correspondences can be used, or pointinteractions can be restricted to local vicinities [28]. In ourQA, we opt for the second solution which allows to use therotational bases (13) and (22) elaborated in Sec. 5. Eventu-ally, the Φ ∈ R(K+1)×(D)(L(1)+L(2)+...+L(N)) matrix en-coding point interactions for point set alignment reads

Φ =[Φ1Φ2 . . .ΦN

], (25)

with Φn, n ∈ {1, . . . , N}, of the formxTn xT

n . . . xTn

−[Q1yn1 ]

T −[Q1yn2 ]

T . . . −[Q1ynL(n)]

T

−[Q2yn1 ]

T −[Q2yn2 ]

T . . . −[Q2ynL(n)]

T

......

. . ....

−[QKyn1 ]T −[QKyn2 ]

T . . . −[QKynL(n)]T

, (26)

with Qk being as in (13) or (22) for the 2D and 3D case,respectively. Φ1, Φ2 and ΦN encode point interactions be-tween every xn and corresponding L(n) � M points ofthe template denoted by superscripted {yn1 ,yn2 , . . . ,ynL(n)}.Note that the latter build N subsets of {y1,y2, . . . ,yM}of different cardinalities L(n), L on average. If differ-ent xn interact with the same ym, the corresponding sub-columns

[[Q1ym]T[Q2ym]T . . . [QKym]T

]Tof Φn can be

computed only once and reused. The final QUBOP for pointset alignment with Φn as in (26) reads

arg minq∈BK+1

qTΦΦTq. (27)

In total, K + 1 = 21 and K + 1 = 81 qubits are re-quired to align point sets on AQC in the 2D and 3D case,respectively. Both transformation estimation and point setalignment need the same number of qubits in the same di-mensions, and the difference lies in the complexity to con-struct P (see Sec. 6.2). Note that if the same template has tobe aligned to multiple references, the corresponding Φ canbe obtained by reusing

[[Q1ym]T[Q2ym]T . . . [QKym]T

]T(which has to be computed only once). The first qubit of qhas to be fixed to |1〉, since the first element of every columncontains a reference point which has to be active during theentire optimisation. The unembedding is performed simi-larly to the case of transformation estimation, see Sec. 5.

6.2. Complexity to Prepare P = ΦΦT

To prepare Φ, O(KDNξ) and O(KDNLξ) operationsare required for the transformation estimation and pointset alignment, respectively. ξ denotes the number of op-erations for multiplying ym with one element of the ad-ditive basis Qk. To obtain the final P, we need to trans-pose Φ and multiply Φ with ΦT which, in the worst case,takes O(K2DN) operations for the transformation estima-tion and O(K2DNL) operations for the point set align-ment. There are also slightly faster algorithms for matrixmultiplication compared to the naıve way [20].

TE K10 20 30 40 50

e2D 0.023 0.026 0.041 0.078 0.17 0.3σ2D 0.012 0.013 0.012 0.012 0.012 0.013eR 0.058 0.062 0.083 0.22 0.47 0.764σR 0.041 0.044 0.041 0.036 0.031 0.03

Table 2: The accuracy of QA under random initial misalignments, for thetransformation estimation (”TE”) and point set alignment (K > 1).

7. Experimental EvaluationThe current generation of D-WAVE annealers does not

support the precision of weights in P necessary for ourmethod [22]3. It is foreseeable that future generations willenable a higher accuracy for couplings. We thus implementand test QA with an AQC sampler on a conventional com-puter (Intel i7-6700K CPU with 32GB RAM). All quanti-tative tests are performed with 21 binary variables corre-sponding to the size of the Q basis in 2D.

We report two error metrics, i.e., the alignment errore2D and the transformation discrepancy eR, together withtheir standard deviations denoted by σ2D and σR, respec-tively. The alignment error e2D =

‖RY−X‖HS‖X‖HS

(‖·‖HS de-notes the Hilbert-Schmidt norm) measures how accuratelythe aligned shape coincides with the reference and requiresground truth correspondences. The transformation discrep-ancy is defined as eR =

∥∥I−RRT∥∥HS , where R is the

recovered rotation. It measures how closely the recoveredtransformation resembles a valid rigid transformation. Theusage of two complementary metrics is necessary becausea low eR does not automatically imply an accurate registra-tion. On the other hand, a low e2D does not quantify howrigid the recovered transformation is.Datasets and Proof of Concept. We use four 2D datasets,i.e., fish [47], qubit, kanji and composer with cardinalitiesvarying from 91 (fish) to 7676 (composer), see Fig. 1 forqualitative registration results. For point sets with up to afew thousand points, the simulation time τP < 1 sec. For∼7.7k, τP grows to 20.178 sec (by a factor of ∼104). Sim-ulation with n = 30 takes already ∼2.5 days. More binaryvariables allow for more elements in the basis Q resultingin more accurate alignment. Note that even with 80 qubits,i.e., for problems with n = 80, annealing on AQC takesaround 100 ms. A simulation with n = 80 is not possibleeven on a conventional supercomputer in a reasonable time.Initial Misalignment and Point Linking. We test how ac-curately our method recovers the transformation under therandom angle of initial misalignment θ and the different sizeof the point linking region. We generate 500 random trans-formations in the range θ ∈ [0; 2π] of the fish dataset andresolve them with QA, for each K ∈ {1, 10, 20, 30}. Theresults are summarised in Table 2. We see that e2D corre-

3the current generation natively supports 9-bit floating-point numbers

7

1 10 19 28 37 46 55 64 73 82 910

0.5

1

1.5

1 10 19 28 37 46 55 64 73 82 910

0.2

0.4

0.6

0.8

1

KA/

K

Colum

n B

Colum

n C

Colum

n D

Col

umn

ECol

umn

FCol

umn

G

Colum

n B

Colum

n C

Col

umn

DCol

umn

ECol

umn

FCol

umn

G

Colum

n B

Colum

n C

Colum

n D

Colum

n E

Colum

n F

Colum

n G

Colum

n B

Colum

n C

Colum

n D

Colum

n E

Colum

n F

Colum

n G

Colum

n B

Colum

n C

Colum

n D

Colum

n E

Colum

n F

Colum

n G

Colum

n B

Col

umn

CCol

umn

DCol

umn

ECol

umn

FCol

umn

G=

00.7

91.5

72.3

63.1

43.9

34.7

1 5.50

0.2

0.4

0.6

0.8

00.7

91.5

72.3

63.1

43.9

34.7

1 5.50

0.1

0.2

0.3

0.4

40 50

Colum

n C

Colum

n D

Colum

n E

Colum

n F

Colum

n G

Colum

n H

Colum

n C

Col

umn

DCol

umn

ECol

umn

FCol

umn

GCol

umn

H

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

10K = 5 15 20

Col

umn

SCol

umn

TC

olum

n U

Colum

n V

Col

umn

SCol

umn

TC

olum

n U

Colum

n V

Col

umn

SCol

umn

TC

olum

n U

Colum

n V

Col

umn

SCol

umn

TC

olum

n U

Colum

n V

10K = 1 20 30

Col

umn

CCol

umn

DCol

umn

ECol

umn

FCol

umn

GCol

umn

H

Colum

n C

Col

umn

DCol

umn

ECol

umn

FCol

umn

GCol

umn

H

Colum

n C

Colum

n D

Colum

n E

Colum

n F

Col

umn

GCol

umn

H

Colum

n C

Col

umn

DCol

umn

ECol

umn

FCol

umn

GCol

umn

HB/ noise ratio, % (template)

noise ratio, % (template)

C/

Figure 3: The metrics as the functions of A/: the size of the point inter-action region parametrised by K; B/: the angle of initial misalignment θ;C/: the template noise ratio.

lates with eR for all tested K. For K = 30 — which cor-responds to one third of the template points — both metricsare still comparably low. We also study how the choice ofthe point interaction region or K affects the accuracy of thetransformation recovery and plot e2D and eR as the func-tions of K for several angles of initial misalignment θ inFig. 3-A. Interacting points are determined with theK near-est neighbour rule for each xn. Recall that according to thesingularity theorem [28], the globally multiply-linked align-ment (here, K = 91) results in a shrinkage of the templateto a single point, which is observed experimentally.

Next, we systematically vary the angle of initial mis-alignment θ in the range [0; 2π] with the angular step π

36and report e2D and eR as the functions of θ, for K ∈{1, 10, 20, 30, 40, 50}. This test reveals the differences inthe transformations caused by θ, which arise due to the com-position and the expressiveness of the chosen basis M , seeFig. 3-B. QA is almost agnostic to θ, which is a desirableproperty of every point set alignment method.Sensitivity to Noise. We systematically add uniformly dis-tributed noise to the template and test the robustness of theproposed QA to outliers in the data, since real data oftencontains outliers. The highest template noise ratio amountsto 50%. Each metric for every noise ratio and every K isaveraged over 50 runs, see Fig. 3-C. σR and σ2D do notexceed 0.057 and 0.03, respectively. We observe both theincreasing alignment error and the discrepancy in the ob-tained transformations with the increasing noise level. Forsmall K, nonetheless, even large noise ratios seem not toinfluence the metrics significantly.Spectral Gap Analysis. Spectral gap ∆(H) is the differ-ence between the energy of the ground state and the second-lowest eigenstate. Each problem has an intrinsic and unique∆(H). Even though a rigorous analysis of the spectral gapis out of the scope of this paper, we make several qualitativeobservations about the energy landscape of QA, the differ-ence in the energy values and the corresponding registra-tions for one exemplary problem. In Fig. 4, we plot the se-

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.01

0.1

1

10

100

Column B

Column C

Column D

1 5 9 13 17 21 25 1 6 11 16 21900

1100

1300

1500

1700

1900

2100

2300

2500

2700

Column B

Column C

Column D

energy-decreasing transitionenergy-decreasing transition

ener

gy

ener

gy

K = 1

Column B

Column C

Column D

Column B

Column C

Column D

Column B

Column C

Column D

K = 30

Figure 4: The sequences of energy-decreasing transitions and the corre-sponding energy values observed in our sampler, for transformation esti-mation (K = 1) and point set alignment with K = 30 interactions perxn. Besides the graphs, we visualise alignment results for selected energyvalues and the angle of initial misalignment θ ∈

{π8, π4, π2

}.

quences of energy-decreasing transitions together with theenergy values in the experiment with fish, for three θ values.We notice that some solutions have very small differencesin the energies and are qualitatively indistinguishable fromeach other. This is accounted for by the choice of the ad-ditive basis, i.e., that the same alignment can be encoded indifferent ways. In contrast, we see significant differencesin the energy values of the qualitatively different solutions(orders of magnitudes larger in the analysed experiment).

We conclude that even though ∆(H) is small, the align-ments corresponding to several few lowest eigenstates arequalitatively similar. This suggests that our selection of thebasis leads to problems with sufficient spectral gaps.

8. Conclusions

This paper introduces AQC for the computer vision com-munity and shows that fundamental low-level problems canbe brought to a representation suitable for solving on AQC.In simulations on a classical computer and in a wide rangeof scenarios, our QA is shown to successfully recover 2Dtransformations which are close approximations of globallyoptimal transformations. With the chosen basis of 20 el-ements, the solutions result in low transformation discrep-ancy and alignment errors. Observations on how to avoidsingularities as well as the noise sensitivity and spectal gapanalysis complement the experimental section.

In future work, our technique can be extended to affinetransformations and other related computer vision prob-lems. We hope to see more research on computer visionmethods with quantum hardware in the next decades.Acknowledgements. This work was supported by the ERC Con-solidator Grant 770784. VG is grateful to Polina Matveeva formany enlightening discussions on the physical foundations of adi-abatic quantum computing. The authors thank Bertram Taetz andHanno Ackermann for reviewing an earlier version of this paper.

8

A. AppendixIn this additional section, we provide details on the se-

lection of the annealing rate, analyse the structure of P andformalise the unembedding, i.e., the conversion of the solu-tion to QUBOP (16) to the solution of the original alignmentproblem on point sets. We preserve the notations referringto the sections and equations from the main matter. Theequations and the figure introduced in this supplement areequipped with Roman numerals.Annealing Rate. Suppose En(s) is the ground state of in-stantaneous Hamiltonian, En(0) is the initial state (groundstate) of the system and Em(s) is any other excited state ofthe instantaneous Hamiltonian. Let s = t

T ∈ [0; 1], whereT is the overall time of interpolation and t is physical time.Then, according to [4], T has to be chosen so that

T � |〈Em(s)|dH/ds|En(s)〉|Enm(s)2

, ∀m 6= n, (i)

where dH/ds is the rate of change of Hamiltonian with re-spect to s and Enm is the difference in the correspondinginstantaneous energies.Analysis of P. Fig. I visualises several exemplary weightmatrices P from the experiments with clean and noisy data(see Sec. 7). There are several observations. First, P =ΦΦT is symmetric upon algorithm design. We also see thatthe columns of Φ can be arbitrarily reshuffled as long asthe correspondences are preserved4. Second, P containsregularly arranged zero submatrices, due to our choice ofthe basis. As soon as a row of Φ induced by qCI, whereCI ∈ {I,−I}, is multiplied by a column of ΦT induced byqCM, where CM ∈ {M,−M}, and vice versa, we obtaina zero entry in P. The reason is that

[I∑i yi]

T [M∑j yj ] = 0

[−I∑i yi]

T [M∑j yj ] = 0

[I∑i yi]

T [−M∑j yj ] = 0

[−I∑i yi]

T [−M∑j yj ] = 0

, (ii)

if∑i yi =

∑j yj , which holds in our case since each row

of Φ except the first row includes all points of Y multi-plied by a single basis element Qk (see Fig. I-(top left) forC pairs resulting in zero matrices). Third, the structure ofP reflects that its diagonal entries encode biases, and non-diagonal elements represent couplings between the qubits.

With the increasing K, the span of the absolute energyvalues increases, due to the higher number of point inter-actions. As expected, P depends on data and the angle ofinitial misalignment between the point sets. For all possi-ble inputs and initial conditions — point sets of differentcardinalities, K and θ — the structure of P is the same for

4a reshuffling of rows requires changing the order of elements in Q

K = 1 K = 20 K = 40

K = 20K = 10 K = 40

A/

B/

, ,

, ,

, ,

, ,

Figure I: Exemplary visualisations of the weight matrix P = ΦΦT inthe experiment with clean (A/) and noisy data with 35% of outliers in thetemplate (B/), for K ∈ {1, 10, 20, 40} and θ ∈

{π4, π}

. The colourscheme and the range of energy values are given to the right of each P.White colour stands for zero entries. The diagonal values in P representbiases (marked in orange on the top left), and non-zero elements representcouplings between the qubits. In the visualisation on the top left, we listthe pairs of C ∈ {I,M,−I,−M} eventually leading to zero matrices.

the chosen basis. From P, we also recognise that the con-sidered alignment problem is not purely combinatorial andrequires high-precision weights Jj,k in (5).Unembedding. Unembedding is the decoding of the solu-tion to QUBOP (16) to the solution of the original alignmentproblem. Upon the design, our QA method assembles theentries of the transformation matrix in the additive basis Qk

(see Secs. (5.1)–(6.1)). Suppose q is the measurement re-sult of q, i.e., it is a classical bitstring with K + 1 elements.Recall that q1 is reserved for reference points and does notcontribute to the assembly of the transformation. Once q ismeasured and returned, we obtain the corresponding trans-formation R by summing up Qk multiplied by qk+1:

R =∑k

qk+1Qk. (iii)

The obtained R is an affine transformation. If the solutionhas to represent a valid rotation matrix Rr, R can be pro-jected to the rotation group by solving the closest orthogo-nal approximation problem with constraints:

min ‖Rr −R‖2HS ,s. t. R−1r = RT

r and det(Rr) = 1.(iv)

For a solution to (iv) by singular value decomposition, see[33].

9

References[1] Steven H. Adachi and Maxwell P. Henderson. Application of

Quantum Annealing to Training of Deep Neural Networks.arXiv e-prints, 2015. 5

[2] Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau,Seth Lloyd, and Oded Regev. Adiabatic quantum computa-tion is equivalent to standard quantum computation. SIAM J.Comput., 37(1):166–194, 2007. 1

[3] Andris Ambainis and Robert Spalek. Quantum algorithmsfor matching and network flows. In STACS, 2006. 5

[4] Mohammad H. S. Amin. Consistency of the adiabatic theo-rem. Physical Review Letters, 102:220401, 2009. 4, 9

[5] Mathieu Aubry, Ulrich Schlickewei, and Daniel Cremers.The wave kernel signature: A quantum mechanical approachto shape analysis. In International Conference on ComputerVision (ICCV) Workshops, pages 1626–1633, 2011. 5

[6] Caglar Aytekin, Serkan Kiranyaz, and Moncef Gabbouj. Au-tomatic object segmentation by quantum cuts. In Interna-tional Conference on Pattern Recognition (ICPR), 2014. 5

[7] Francisco Barahona. On the computational complexity ofising spin glass models. Journal of Physics A: Mathematicaland General, 15(10):3241–3253, 1982. 4

[8] Paul Benioff. The computer as a physical system: A micro-scopic quantum mechanical hamiltonian model of comput-ers as represented by turing machines. Journal of StatisticalPhysics, 22(5):563–591, 1980. 1

[9] Charles H. Bennett, Francois Bessette, Gilles Brassard,Louis Salvail, and John Smolin. Experimental quantumcryptography. J. Cryptol., 5(1):3–28, 1992. 5

[10] Charles H. Bennett and Gilles Brassard. Quantum cryptog-raphy: Public key distribution and coin tossing. In Interna-tional Conference on Computers, Systems, and Signal Pro-cessing, 1984. 5

[11] Aleta Berk Finnila, Maria Gomez, C. Sebenik, CatherineStenson, and Jimmie D. Doll. Quantum annealing: A newmethod for minimizing multidimensional functions. Chemi-cal Physics Letters, 219, 1994. 3

[12] Ethan Bernstein and Umesh Vazirani. Quantum complex-ity theory. In Symposium on Theory of Computing (STOC),1993. 5

[13] Paul J. Besl and Neil D. McKay. A method for registration of3-d shapes. Transactions on Pattern Analysis and MachineIntelligence (TPAMI), 14(2):239–256, 1992. 2, 6

[14] Zhengbing Bian, Fabian Chudak, Robert Brian Israel, BradLackey, William G. Macready, and Aidan Roy. Mappingconstrained optimization problems to quantum annealingwith application to fault diagnosis. Frontiers in ICT, 3:14,2016. 1

[15] Zhengbing Bian, Fabian Chudak, William G. MacReady,Lane Clark, and Frank Gaitan. Experimental determinationof ramsey numbers with quantum annealing. Physical Re-view Letters, 111(13):130505, 2013. 1

[16] Tomas Boothby, Andrew D. King, and Aidan Roy. Fastclique minor generation in chimera qubit connectivitygraphs. Quantum Information Processing, 15(1):495–508,2016. 4

[17] Max Born and Vladimir Fock. Beweis des adiabatensatzes.Zeitschrift fur Physik, 51(3):165–180, 1928. 1, 3, 4

[18] Edward Boyda, Saikat Basu, Sangram Ganguly, AndrewMichaelis, Supratik Mukhopadhyay, and Ramakrishna R.Nemani. Deploying a quantum annealing processor to de-tect tree cover in aerial imagery of california. PLOS ONE,12(2), 2017. 5

[19] Patrick J. Coles et al. Quantum Algorithm Implementationsfor Beginners. arXiv e-prints, 2018. 1

[20] Don Coppersmith and Shmuel Winograd. Matrix multiplica-tion via arithmetic progressions. Journal of Symbolic Com-putation, 9(3):251 – 280, 1990. 7

[21] D-Wave Systems. D-Wave Announces D-Wave 2000QQuantum Computer and First System Order. https://www.dwavesys.com/press-releases/..., 2019.online; accessed 15 October 2019. 1

[22] D-Wave Systems. Technical Description of the D-WaveQuantum Processing Unit. https://docs.dwavesys.com/docs/latest/doc_qpu.html, 2019. online; ac-cessed 5 November 2019. 4, 7

[23] Vasil S. Denchev, Sergio Boixo, Sergei V. Isakov, Nan Ding,Ryan Babbush, Vadim Smelyanskiy, John Martinis, andHartmut Neven. What is the computational value of finite-range tunneling? Phys. Rev. X, 6:031015, 2016. 1

[24] David Deutsch and Richard Jozsa. Rapid solution of prob-lems by quantum computation. Proceedings of the RoyalSociety of London Series A, 439(1907):553–558, 1992. 5

[25] Simon J. Devitt. Performing quantum computing experi-ments in the cloud. Phys. Rev. A, 94:032329, 2016. 1

[26] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Joshua La-pan, Andrew Lundgren, and Daniel Preda. A quantum adia-batic evolution algorithm applied to random instances of annp-complete problem. Science, 292(5516):472–475, 2001.1, 2, 3, 4

[27] Richard P. Feynman. Simulating physics with computers. In-ternational Journal of Theoretical Physics, 21(6):467–488,1982. 1, 4, 5

[28] Vladislav Golyanik and Christian Theobalt. Optimising forscale in globally multiply-linked gravitational point set reg-istration leads to singularities. In International Conferenceon 3D Vision (3DV), 2019. 6, 7, 8

[29] Vladislav Golyanik, Christian Theobalt, and Didier Stricker.Accelerated gravitational point set alignment with alteredphysical laws. In International Conference on Computer Vi-sion (ICCV), 2019. 2, 6

[30] Lov K. Grover. A fast quantum mechanical algorithm fordatabase search. In Annual ACM Symposium on Theory ofComputing, pages 212–219, 1996. 1, 2, 5

[31] Kuk-Hyun Han and Jong-Hwan Kim. Genetic quantum algo-rithm and its application to combinatorial optimization prob-lem. In Congress on Evolutionary Computation, 2000. 5

[32] Kuk-Hyun Han and Jong-Hwan Kim. Quantum-inspiredevolutionary algorithm for a class of combinatorial optimiza-tion. Trans. Evol. Comp, 6(6):580–593, 2002. 5

[33] Nicholas J. Higham. Matrix nearness problems and applica-tions. Applications of Matrix Theory, 22, 1989. 9

10

[34] Berthold K. P. Horn, Hugh M. Hilden, and Shahriar Negah-daripour. Closed-form solution of absolute orientation usingorthonormal matrices. J. Opt. Soc. Am. A, 5(7):1127–1135,1988. 2

[35] Ernst Ising. Beitrag zur theorie des ferromagnetismus.Zeitschrift fur Physik, 31(1):253–258, 1925. 3

[36] Tadashi Kadowaki and Hidetoshi Nishimori. Quantum an-nealing in the transverse ising model. Phys. Rev. E, 58:5355–5363, 1998. 3

[37] Bruce E. Kane. A silicon-based nuclear spin quantum com-puter. Nature, 393:133137, 1998. 2

[38] Amir Khoshaman, Walter Vinci, Brandon Denis, EvgenyAndriyash, Hossein Sadeghi, and Mohammad H. Amin.Quantum variational autoencoder. Quantum Science andTechnology, 4(1), 2018. 5

[39] Scott Kirkpatrick, C. Daniel Gelatt, and Mario P. Vecchi. Op-timization by simulated annealing. Science, 220(4598):671–680, 1983. 2, 3

[40] Trevor Lanting et al. Entanglement in a quantum annealingprocessor. Phys. Rev. X, 4:021041, 2014. 1

[41] Kenneth Levenberg. A method for the solution of certainnon-linear problems in least squares. Quarterly Journal ofApplied Mathmatics, II(2):164–168, 1944. 6

[42] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quan-tum principal component analysis. Nature Physics, 10, 2013.1, 2, 5

[43] Yuri Manin. Computable and Noncomputable. Sov. Radio.,1980. 1, 4

[44] Donald W. Marquardt. An algorithm for least-squares esti-mation of nonlinear parameters. SIAM Journal on AppliedMathematics, 11(2):431–441, 1963. 6

[45] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N.Rosenbluth, Augusta H. Teller, and Edward Teller. Equa-tion of state calculations by fast computing machines. TheJournal of Chemical Physics, 21(6):1087–1092, 1953. 3

[46] Andrea Morello et al. Single-shot readout of an electron spinin silicon. Nature, 467:687691, 2010. 2

[47] Andriy Myronenko and Xubo Song. Point-set registration:Coherent point drift. Transactions on Pattern Analysis andMachine Intelligence (TPAMI), 2010. 1, 2, 7

[48] Kae Nemoto, Michael Trupke, Simon J. Devitt, Ashley M.Stephens, Burkhard Scharfenberger, Kathrin Buczak, To-bias Nobauer, Mark S. Everitt, Jorg Schmiedmayer, andWilliam J. Munro. Photonic architecture for scalable quan-tum information processing in diamond. Phys. Rev. X,4:031022, 2014. 1

[49] Florian Neukart, Gabriele Compostella, Christian Seidel,David von Dollen, Sheir Yarkoni, and Bob Parney. Trafficflow optimization using a quantum annealer. Frontiers inICT, 4:29, 2017. 1, 2

[50] Hartmut Neven, Vasil S. Denchev, Geordie Rose, andWilliam G. Macready. Qboost: Large scale classifier trainingwith adiabatic quantum optimization. In Asian Conferenceon Machine Learning (ACML), 2012. 5

[51] Hartmut Neven, Geordie Rose, and William G. Macready.Image recognition with an adiabatic quantum computer I.Mapping to quadratic unconstrained binary optimization.arXiv e-prints, 2008. 5

[52] Nga T. T. Nguyen and Garrett T. Kenyon. Image classifi-cation using quantum inference on the D-Wave 2X. arXive-prints, 2019. 5

[53] Michael A. Nielsen and Isaac L. Chuang. Quantum Compu-tation and Quantum Information: 10th Anniversary Edition.Cambridge University Press, 2011. 2, 4

[54] Daniel O’Malley, Velimir V. Vesselinov, Boian S. Alexan-drov, and Ludmil B. Alexandrov. Nonnegative/binary matrixfactorization with a d-wave quantum annealer. PLOS ONE,13, 2018. 5

[55] Ojas Parekh, Jeremy Wendt, Luke Shulenburger, AndrewLand ahl, Jonathan Moussa, and John Aidun. Benchmark-ing Adiabatic Quantum Optimization for Complex NetworkAnalysis. arXiv e-prints, 2016. 4

[56] James L. Park. The concept of transition in quantum me-chanics. Foundations of Physics, 1(1):23–33, 1970. 5

[57] Rudolph Peierls. On ising’s model of ferromagnetism. Math-ematical Proceedings of the Cambridge Philosophical Soci-ety, 32(3):477–481, 1936. 3

[58] Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd. Quan-tum support vector machine for big data classification. Phys-ical Review Letters, 113, 07 2013. 1, 5

[59] Eleanor Rieffel and Wolfgang Polak. An introduction toquantum computing for non-physicists. ACM ComputingSurveys (CSUR), 32(3):300–335, 2000. 1, 2

[60] Tobias Schmitt-Manderbach, Henning Weier, Martin Furst,Rupert Ursin, Felix Tiefenbacher, Thomas Scheidl, JosepPerdigues, Zoran Sodnik, Christian Kurtsiefer, John G. Rar-ity, Anton Zeilinger, and Harald Weinfurter. Experimentaldemonstration of free-space decoy-state quantum key dis-tribution over 144 km. In European Conference on Lasersand Electro-Optics and the International Quantum Electron-ics Conference, 2007. 5

[61] Peter W. Shor. Polynomial-time algorithms for prime fac-torization and discrete logarithms on a quantum computer.SIAM J. Comput., 26(5):1484–1509, 1997. 1, 2, 5

[62] Peter W. Shor. Introduction to Quantum Algorithms. arXive-prints, 2001. 4

[63] Janus H. Wesenberg, Arzhang Ardavan, G. Andrew D.Briggs, John J. L. Morton, Robert J. Schoelkopf, David I.Schuster, and Klaus Mølmer. Quantum computing with anelectron spin ensemble. Phys. R. Lett., 103:070502, 2009. 2

[64] William K. Wootters and Wojciech Hubert Zurek. A singlequantum cannot be cloned. Nature, 299(5886), 1982. 5

[65] Richard Y. Li, Rosa Di Felice, Remo Rohs, and Daniel Li-dar. Quantum annealing versus classical machine learningapplied to a simplified computational biology problem. npjQuantum Information, 4:14, 02 2018. 1, 2

[66] Jiaolong Yang, Hongdong Li, Dylan Campbell, and YundeJia. Go-icp: A globally optimal solution to 3d icp point-setregistration. Transactions on Pattern Analysis and MachineIntelligence (TPAMI), 38(11):2241–2254, 2016. 2

[67] Floris A. Zwanenburg, Andrew S. Dzurak, Andrea Morello,Michelle Y. Simmons, Lloyd C. L. Hollenberg, GerhardKlimeck, Sven Rogge, Susan N. Coppersmith, and Mark A.Eriksson. Silicon quantum electronics. Rev. Mod. Phys.,85:961–1019, 2013. 2

11